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User blog:Ecl1psed276/Star Notation Revamped! - Introduction and Analysis
Well, I made a blog post a few months back about my new notation "Star notation". I've decided to overhaul this notation and extend it to be really powerful. Part 1 - Bracket notation This part is really simple, so it will be brief. Bracket notation consists of a,b;c where a, b, and c are numbers. \(a\) is called the base, and \(b\) is called the key. There will always be exactly 2 entries before the semicolon (even when we get to the higher parts of this notation). The rules are as follows: #\(a,1;c = a\) #\(a,b;0 = a,b = a*b\) #\(a,b;c = [a,(a,b-1;c);c-1] \) With these rules, a,b;0=a,b is equivalent to multiplying a and b, a,b;1 equals a^b, a,b;2 equals a tetrated to b, and so on. In general, a,b;c equals \(a\uparrow\uparrow...\uparrow c\) with b arrows. Thus, a,b;1 is approximately \(f_2(b)\), a,b;2 is approximately \(f_3(b)\), and so on. The ordinal limit of this notation is therefore \(\omega\). Part 2 - Linear array notation In this section, the notation changes to allow arrays of numbers in the square brackets. It looks like \(b,k;n_1,n_2...n_m \). The rules for evaluating arrays are as follows: (note that you can remove trailing zeros) #Key rule: \(b,1;n_1,n_2...n_m = b\) #Base rule: \(b,k;0 = b*k\) #Successor rule: \(b,k;n_1+1,n_2...n_m = [b,b,k-1;n_1+1,n_2...n_m;n_1,n_2...n_m]\) #Expansion rule: \(b,k;0,0...0,n_1+1,n_2...n_m = b,k;0,0...k,n_1,n_2...n_m\) According to rule 3, a,b;0,1 will expand to a,b;b, which has a growth rate of \(f_\omega\), as per the previous section. a,b;1,1 will expand to [a,(a1,1b-1);0,1], which will eventually become a0,1a0,1a...a0,1a with b a's. This has a growth rate of \(f_{\omega+1}\). Thus, \(3,64;1,1 \approx G_{64}\) where \(G_{64}\) is Graham's number. Continuing, we find that a,b;2,1 has a growth rate of \(f_{\omega+2}\), a,b;3,1 has a growth rate of \(f_{\omega+3}\), and a,b;0,2 = a,b;b,1 has a growth rate of \(f_{\omega2}\). We also have the following comparisons: 1,2 corresponds to \(\omega2+1\). 2,2 corresponds to \(\omega2+2\). 0,3 corresponds to \(\omega3\). 0,0,1 corresponds to \(\omega^2\). 1,0,1 corresponds to \(\omega^2+1\). 0,1,1 corresponds to \(\omega^2+\omega\). 0,0,2 corresponds to \(\omega^22\). 0,0,0,1 corresponds to \(\omega^3\). 0,0,0,0,1 corresponds to \(\omega^4\). Thus, the ordinal limit of linear array notation is \(\omega^\omega\). Part 3 - Nested array notation In Nested array notation, we will introduce the new concept of a separator. A separator looks like \(S_1 n_2 S_2 ... n_{m-1} S_{m-1} n_m\) where \(n_i\) are numbers, and \(S_i\) are also separators. The comma that we've seen in linear array notation is just a shorthand for 0. We can also have 1, 2, 0,1, 0,0,1, [011], etc... A valid array in Nested array notation could look something like this: [99,99;0[21,45,1]341]. The next separator after the comma is 1. [011] expands to 0,0,0...0,1 with k commas. Then, we have [111], [0,111], [0,0,111]... etc. And then [012] expands to [0,0,0...0,111] with k commas. We can also have [013], [010,1], [01011], [0101011]... etc. Then, [021] expands to [0101011] with k 1's. Further, we can have [031], [00,11], [01,11], [00,21], [00,0,11], [0[011]1], , [0[0[011]1]1], and so on... which forms the limit of nested array notation. An ordinal notation We can slightly modify this notation to form an ordinal notation, instead of a large number notation. The large number notation is \(a,b;C\), but for the ordinal notation, we will remove the numbers \(a\) and \(b\), and only worry about the array \(C\). The rules for the ordinal notation are all identical to the number notation, except now we don't need the base rule and key rule anymore. Analysis The following is a table of arrays with their ordinal correspondences. Thus, the ordinal limit of nested array notation is \(\varepsilon_0\). Part 4 - Hyper-nested array notation For this part, we will introduce a new separator, the double comma (,,). It is a new, high-level separator. When we are evaulating an array and we see a double comma, we have to look outward until we find a regular [] separator. This is analogous to standard OCFs, because if you see a \(\Omega\) in an OCF, you have to look outward until you find a \(\psi\) function to diagonalize it. Here are some examples using the double comma: 0,,1 expands to [0[0[0...001]1]...1]1] [00,,11,,1] expands to [0[0[0...001]1]...1]1,,1] 0,,2 expands to [0[0[0...001,,1]1,,1]...1,,1]1,,1] 0,,0,,1 expands to [0,,1[0,,1[0...0,,101]1]...1]1] Then, [0,,11] will expand to 0,,0,,0...0,,1, just as [011] expands to 0,0,0...0,1. If you have a standard separator a, you can also think of it as ,a with a comma in front. The separator ,,a will expand into double commas, just as ,a expands into single commas. Analysis (up to \(\varphi(\omega,0)\)) I didn't have to make two separate tables here, but I decided to just because they were getting kinda long :/ More Analysis (up to the BHO) We have now reached the limit of double commas, which happens to be the Bachmann-Howard ordinal. The next obvious extension is the triple comma. When we see a triple comma, we have to look outward until we find a separator of lower level than the triple comma (i.e. a regular seperator [] or a double-comma-seperator ,,[]). Then, the expression expands in much the same way as with double commas. We can further introduce quadruple commas, quintuple commas, and so on, for as many positive integers as there are (there are a lot of those). Even More Analysis (up to \psi(\Omega_\omega)) This is the end of Hyper-Nested Array Notation. In the next part, we will introduce a new kind of separator that generalizes the multiple commas: the curly brackets {}. Part 5 - Pre-star notation Now we will learn about curly brackets. The comma is a shorthand for {0}, the double comma is {1}, then we can have {2}, {3}.... and so on. So the array [0,,,01] can also be expressed as [0{2}01]. Note that the seperator {2}0 acts like a single seperator, a curly bracket seperator combined with a square bracket seperator. We can also have {0,1} which just expands to {n}. Then, we can have {1,1}, {0,2}, {0,0,1}, {00,,11}, {00{0,1}11}, {0,,1}, {0{0,1}1}.... and so on. If you have been following so far, this part shouldn't be too hard to understand. Part 6 is where things get weird. Analysis The limit of Pre-Star Notation is \(\psi(\psi_I(0))\). Part 6 - Single star notation This is the part where star notation starts to get weird. This part will reach the limit of pDAN. WIP. Part 7 - Double star notation This part will reach an unknown ordinal level, but it will go past Hyp cos's pDAN and probably DAN. I might merge this with part 8, not sure yet. WIP. Part 8 - Full star notation This part will reach past DAN, and its ordinal limit will probably be (0,0,0)(1,1,1)(2,2,2) in BMS. WIP. Category:Blog posts